Law of sines

In trigonometry, the sine theorem establishes a relationship between the angles of a general triangle and the opposite sides.

Definition

A, b and c, the sides of a triangle, α, β and γ, respectively, the opposite angle, and r is the radius of the circumcircle, the following applies to the sine function:

If angle is to be calculated in the triangle using the sine theorem, it must be ensured that it is in the interval [0 °, 180 °] are in general two different angles with the same sine value; this ambiguity is that of the Kongruenzsatzes SSW.

For relationship with the Kongruenzsätzen and the systematics of the triangle calculation, see the article on the law of cosines.

In spherical trigonometry, there is a corresponding set, which is also called the law of sines.

History

He was Abu Nasr Mansur ( Persian mathematician and astronomer; around 960-1036 AD) demonstrated for the first time. The first proof is attributed in a few sources Al- Battani, in other Abu Mahmud al - Chudschandi.

Evidence

The drawn amount divided the triangle into two right triangles part, in which one can express the values ​​of sine and each as the ratio of the opposite side to the hypotenuse and:

Solving for yields:

Equating is accordingly obtained

Dividing by now, we get the first part of the assertion:

The equality with results accordingly by using the height or. In order also to show compliance with that, strictly speaking, not part of the law of sines, ( peripheral angle) and central angle you need the well-known theorems on circumferential angle ( central angle ).

Example of use

The following numerical values ​​are rough approximations. In a triangle ABC the following pages and angle parameters are known ( designations as usual):

We are looking for the sizes of the remaining sides and angles. First, you used the law of sines to calculate. After that applies

What to can be worked

Resulting in, using the inverse sine, the inverse function of the sine

Can be calculated.

Actually there is a second angle with the same sine value, namely. This comes as a solution but not considered, since otherwise the sum of the angles of the triangle would exceed prescribed.

Is now obtained using the angle sum

The page length should be re- calculated using the law of sines. ( Also, the law of cosines would be possible here.) It is

By rearranging leads to the result